On Whitney embedding of o-minimal manifolds

Abstract

We prove a definable version of the Whitney embedding theorem for abstract-definable ${\mathcal{C}}^p$ manifolds with $1\le p<\infty$, namely: every abstract-definable ${\mathcal{C}}^p$ manifold is abstract-definable $C^p$ embedded into $R^N$, for some positive integer $N$. As a consequence, we show that every abstract-definable ${\mathcal{C}}^p$ manifold has a compatible ${\mathcal{C}}^{p+1}$ atlas.

10 pages